4.2: The SIS Model

The SI model may be extended to the SIS model, where an infective can recover and become susceptible again. We assume that the probability that an infective recovers during time \(\Delta t\) is given by \(\gamma \Delta t\). Then the total number of infective people that recover during time \(\Delta t\) is given by \(I \times \gamma \Delta t\), and

\[I(t+\Delta t)=I(t)+\beta \Delta t S(t) I(t)-\gamma \Delta t I(t) \nonumber \]

or as \(\Delta t \rightarrow 0\)

\[\frac{d I}{d t}=\beta S I-\gamma I \nonumber \]

which we diagram as

\[S \underset{\gamma I}{\stackrel{\beta S I}{\rightleftharpoons}} I \text {. } \nonumber \]

Using \(S+I=N\), we eliminate \(S\) from (4.2.2) to obtain

\[\frac{d I}{d t}=(\beta N-\gamma) I\left(1-\frac{\beta}{\beta N-\gamma} I\right) \nonumber \]

which is again a logistic equation, but now with growth rate \(\beta N-\gamma\) and carrying capacity \(N-\gamma / \beta\). In the SIS model, an epidemic will occur if \(\beta N>\gamma\). And if an epidemic does occur, then the disease becomes endemic with the number of infectives at equilibrium given by \(I_{*}=N-\gamma / \beta\), and the number of susceptibles given by \(S_{*}=\gamma / \beta\).

In general, an important metric for whether or not an epidemic will occur is called the basic reproductive ratio. The basic reproductive ratio is defined as the expected number of people that a single infective will infect in an otherwise susceptible population. To compute the basic reproductive ratio, define \(l(t)\) to be the probability that an individual initially infected at \(t=0\) is still infective at time \(t\). Since the probability of being infective at time \(t+\Delta t\) is equal to the probability of being infective at time \(t\) multiplied by the probability of not recovering during time \(\Delta t\), we have

\[l(t+\Delta t)=l(t)(1-\gamma \Delta t) \nonumber \]

or as \(\Delta t \rightarrow 0\)

\[\frac{d l}{d t}=-\gamma l \nonumber \]

With initial condition \(l(0)=1\)

\[l(t)=e^{-\gamma t} \nonumber \]

Now, the expected number of secondary infections produced by a single primary infective over the time period \((t, t+\Delta t)\) is given by the probability that the primary infective is still infectious at time \(t\) multiplied by the expected number of secondary infections produced by a single infective during time \(\Delta t\); that is, \(l(t) \times S(t) \beta \Delta t\). Here, the definition of the basic reproductive ratio assumes that the entire population is susceptible so that \(S(t)=N\). Therefore, the expected number of secondary infectives produced by a single primary infective in a completely susceptible population is

\[\begin{aligned} \int_{0}^{\infty} \beta l(t) N d t &=\beta N \int_{0}^{\infty} e^{-\gamma t} d t \\[4pt] &=\frac{\beta N}{\gamma} \end{aligned} \nonumber \]

The basic reproductive ratio, written as \(\mathcal{R}_{0}\), is therefore defined as

\[\mathcal{R}_{0}=\frac{\beta N}{\gamma} \nonumber \]

and from (4.2.4), we can see that in the SIS model an epidemic will occur if \(R_{0}>1\). In other words, an epidemic can occur if an infected individual in an otherwise susceptible population will on average infect more than one other individual.

We have also seen an analogous definition of the basic reproductive ratio in our previous discussion of age-structured populations \((\S 2.5)\). There, the basic reproductive ratio was the number of female offspring expected from a new born female over her lifetime; the population size would grow if this value was greater than unity.

In the SIS model, after an epidemic occurs the population reaches an equilibrium between susceptible and infective individuals. The effective basic reproductive ratio of this steady-state population can be defined as \(\beta S_{*} / \gamma\), and with \(S_{*}=\gamma / \beta\) this ratio is evidently unity. Clearly, for a population to be in equilibrium, an infective individual must infect on average one other individual before he or she recovers.